Date of Project
3-28-2025
Document Type
Honors Thesis
School Name
College of Arts and Sciences
Department
Mathematics
Major Advisor
Dr. Gregory Kelsey
Second Advisor
Dr. Susan White
Abstract
In a real vector space, a symmetric convex body containing the origin and compact under the Euclidean norm uniquely defines a norm via the Minkowski functional, where the closed unit ball of this norm is the convex body. This establishes a one-to-one correspondence between convex bodies and norms. The norm derived from the convex body allows for the definition of a metric, enabling the study of isometries within the space. However, the symmetries of the convex body itself do not completely determine the isometries of the associated Minkowski space. Instead, the symmetries of the most symmetric shape a convex body can be linearly deformed into correspond to Euclidean and possibly newly defined non-Euclidean isometries of the space. This thesis provides an accessible review of the connection between convex bodies and norms via the Minkowski functional and explores how the symmetries of convex bodies help predict the Euclidean and non-Euclidean isometries of the associated Minkowski space.
Recommended Citation
Kertesz, Fanni, "Circles Are So Euclidean: Exploring Convex Bodies, Norms, Symmetries, and Isometries via the Minkowski Functional" (2025). Undergraduate Theses. 165.
https://scholarworks.bellarmine.edu/ugrad_theses/165